Hamiltonian systems with boundaries

TL;DR

This paper demonstrates how boundary conditions in Hamiltonian systems can be formulated as constraints, providing a purely algebraic approach applicable to various models with boundaries, including string theory and electrodynamics.

Contribution

It introduces an algebraic method to derive boundary conditions as Hamiltonian constraints, avoiding reliance on Lagrangian formulations, and constructs boundary Dirac brackets for different field theories.

Findings

Boundary conditions can be derived as Hamiltonian constraints.

Constructed boundary Dirac brackets for multiple models.

Discovered a novel result for electrodynamics with boundaries.

Abstract

Lately, to provide a solid ground for quantization of the open string theory with a constant B-field, it has been proposed to treat the boundary conditions as hamiltonian constraints. It seems that this proposal is quite general and should be applicable to a wide range of models defined on manifolds with boundaries. The goal of the present paper is to show how the boundary conditions can arise as constraints in a purely algebraic fashion within the Hamiltonian approach without any reference to the Lagrangian formulation of the theory. The construction of the boundary Dirac brackets is also given and some subtleties are pointed out. We consider four examples of field theories with boundaries: the topological sigma model, the open string theory with and without a constant $B$-field and electrodynamics with topological term. A curious result for electrodynamics on a manifold with boundaries is presented.